2 PARTICULAR CASES AND EXAMPLES 251 and that there exists a positive nonzero G-invariant measure /u- on X. Let P (resp. C) be a G -packing (resp. a G-covering) such that P and C are /n-integrable. Show that ft(Q £ /A(P). (Remark that ^t(C) ^ £ p(C n j • P) = £ /^- r - C n P).) seG se G (b) Suppose that there exists a G-invariant distance d on X defining the topology of X. Let A(G) denote the greatest lower bound of the numbers /u,(C), where C runs through all integrable G-coverings of X. Let r be a real number >0 such that there exists a point a e X for which ju,(B(#; r)) > A(G). Show that there exists s ^ e in G such that d(a, s - a) < 2r. (c) Suppose that X is a locally compact group, /x a left Haar measure on X, and G a denumerable subgroup acting on X by left translations. Show that if A is an integrable subset of X such that fju(A) > A(G), then there exists s e G n AA"1 such that s ^ e. (d) With the same hypotheses as in (a), let F be a jie-integrable G -tessellation and let Go be a subgroup of finite index h in G. If si9 . . . , sh are a system of representatives of the right cosets of GO in G, show that F0 — (J St • F is a Go-tessellation. l€f<lr (e) Same hypotheses as in (a). Let/^ 0 be a /u,-integrable function on X. Show that there exist two points a, b in X such that f /(*) dp,(x) and ^(P) £ f(s - b) £ f /( JX see Jx (*) (Observe that if g ^ 0 is an integrable function and E an integrable subset of X, then there exists c e E such that g(x) dp,(x) ^ #(c)/x(E), and c' G E such that 2. PARTICULAR CASES AND EXAMPLES (14.2.1) On the additive group R, Lebesgue measure (13.1.4) is a Haar measure (left and right, since R is commutative). This follows from the formula for change of variable (8.7.4) applied to the function (p(£) = £ + a, which gives f + °°/(f + a) dt = f+ °°/(0 dt for all/e Jfc(R) and all a e R. J — oo J -~ oo (14.2.2) Now consider the multiplicative group R* of real numbers >0. This is a locally compact commutative group ((13.18.4), (4.1.2), and (4.1.4)) For each function fe JTC(R*), there exists a compact interval [a, b] with 0 < a < b, containing the support of /. Hence for each interval [c, d] in R* containing Supp(/), the integral J (/(O dt)/t is defined and its value, r 4* oo which we denote by (/(O dt)/t, is independent of the choice of interval /* + OO [c, d] containing Supp(/). We assert that /K»| (f(t) dt)ft is a Haar measure on R *. From the formula for change of variable (8.7.4), it follows that, for each s > 0, D in R such that